Longitudinal oscillations for eigenfunctions in rod like structures

TL;DR

This work develops a rigorous spectral asymptotic framework for the Laplacian in thin rod-like domains $G_\\u0005varepsilon$ with cross-sections scaled by $\\varepsilon$ and mixed boundary conditions (Dirichlet on end faces, Neumann on the sides). The authors derive a reduced one-dimensional limit model that preserves geometric information through the cross-section area weight $|D_{x_1}|$ and prove convergence of eigenvalues and eigenfunctions: $\\lambda_n^{\\u0005varepsilon}\\to\\lambda_n^0$ and $u_n^{\\u0005varepsilon}\\to U_n^0$ in $H^1$, with the limit problem $-\\partial_{x_1}(|D_{x_1}|\\partial_{x_1}U^0)=\\lambda^0 |D_{x_1}|U^0$ on $(\\ell_0,\\ell_1)$ and Dirichlet ends. They provide explicit and numerical results for prism-like geometries, showing that low-frequency modes are predominantly longitudinal, while observing that transverse oscillations require high-frequency components; the appendix extends the analysis to Neumann and Dirichlet variants. Overall, the paper delivers a rigorous dimensional reduction for slender structures and offers practical insights for diffusion or vibration analysis in engineering applications.

Abstract

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the lateral boundary. We focus on the low frequencies. We study the asymptotic behavior of the eigenvalues and associated eigenfunctions, which are approached as $\varepsilon\to 0$ by those of a 1D model with Dirichlet boundary conditions, but which takes into account the geometry of the domain. Explicit and numerical computations enlighten the interest of this study, when the parameter becomes smaller. At the same time they show that in order to capture oscillations in the transverse direction we need to deal with the high frequencies. For prism like domains, we show the different asymptotic behavior of the spectrum depending on the boundary conditions.

Figures (10)

• Figure 1: Approximations of eigenfunctions of \ref{['eq:ben4']} with $G_\varepsilon=(0,1)\times (0,\varepsilon)\times (0,\varepsilon)$ and $\varepsilon=0.1$. The figures are obtained choosing the eigenvalues $\lambda_1^\varepsilon=\pi^2\approx 9.87,$$\lambda_7^\varepsilon=49\pi^2\approx 484.08$ and $\lambda_{11}^\varepsilon=101\pi^2\approx 1002.85.$
• Figure 2: Approximations of eigenfunctions of \ref{['eq:ben4']} with $G_\varepsilon=(0,2^{-1})\times(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon) \, \cup \, \{2^{-1}\}\times (-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon) \, \cup \, (1/2,1)\times (-\varepsilon2^{-1},\varepsilon2^{-1})\times (-\varepsilon2^{-1},\varepsilon2^{-1})$ and $\varepsilon=8^{-1}$. The figures are obtained choosing the eigenvalues $\lambda_1^\varepsilon \approx 9.87,$$\lambda_7^\varepsilon\approx 247.50$ and $\lambda_{11}^\varepsilon\approx 332.88.$
• Figure 3: Approximations of eigenfunctions of \ref{['eq:ben4']} with $G_\varepsilon=(0,1)\times (0,\varepsilon)\times (-\varepsilon,\varepsilon h(x_1))$ for a certain function $h$ satisfying \ref{['eq:ben13']} and $\varepsilon=0.1$. The figures are obtained choosing the eigenvalues $\lambda_1^\varepsilon\approx 7.85,$$\lambda_5^\varepsilon\approx 236.43$ and $\lambda_7^\varepsilon\approx 326.39$.
• Figure 4: Approximations of eigenfunctions of \ref{['eq:ben25']} with $G_\varepsilon=(0,1)\times (0,\varepsilon)\times (0,\varepsilon)$ and $\varepsilon=0.1$. The figures are obtained choosing the eigenvalues $\lambda_1^\varepsilon=\pi^2\approx 9.87,$$\lambda_7^\varepsilon=49\pi^2\approx 484.09$ and $\lambda_{11}^\varepsilon=100\pi^2\approx 992.65.$
• Figure 5: Approximations of eigenfunctions of \ref{['eq:ben25']} with $G_\varepsilon=(0,2^{-1})\times(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon) \, \cup \, \{2^{-1}\}\times (-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon) \, \cup \, (1/2,1)\times (-\varepsilon2^{-1},\varepsilon2^{-1})\times (-\varepsilon2^{-1},\varepsilon2^{-1})$ and $\varepsilon=8^{-1}$. The figures are obtained choosing the eigenvalues $\lambda_1^\varepsilon \approx 9.35,$$\lambda_4^\varepsilon\approx 158.11$, $\lambda_5^\varepsilon\approx 160.85$.

Theorems & Definitions (3)

• Lemma 1
• proof
• Theorem 2